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Some fixed point generalizations are not real generalizations. (English) Zbl 1251.54045

The authors show how several recent results in metric fixed point theory which appear as generalizations of previous results, can in fact readily be deduced as corollaries of the latter.

This concerns mainly contractive conditions, where some expression such as $d\left(fx,fy\right)\le k\phantom{\rule{0.166667em}{0ex}}d\left(x,y\right)$ is replaced by something like $d\left(fx,fy\right)\le k\phantom{\rule{0.166667em}{0ex}}d\left(gx,gy\right)$. While the latter condition is more general, one can return to the first case by considering a map $h$ defined on the range of $g$ by $h\left(gx\right)=fx$.

The authors apply this transformation to derive a host of results concerning contractive mappings in cone metric spaces, multivalued contractions in complete metric spaces, and nonexpansive mappings between metric spaces and Banach spaces.

Reviewer’s remarks: For cone metric spaces, some of the results mentioned in the paper under review can be reduced even further to the well-known situation in metric spaces, cf. S. Janković, Z. Kadelburg and S. Radenović [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 7, 2591–2601 (2011; Zbl 1221.54059)]. In this regard, note also that Y.-Q. Feng and W. Mao [Fixed Point Theory 11, No. 2, 259–264 (2010; Zbl 1221.54055)] proved that, for a cone metric space $X$ with cone metric $d$ and cone $P$ taking values in a Banach space, $D\left(x,y\right)=inf\left\{\parallel u\parallel :u\in P,\phantom{\rule{4pt}{0ex}}u\ge d\left(x,y\right)\right\}$ defines a metric on $X$ so that $\left(X,D\right)$ is complete if and only if $\left(X,d\right)$ is complete.

By taking into account such reductions, authors and referees should in principle be able to improve the overall significance of research papers in this field. Unfortunately, it appears unlikely that the increase of superfluous publications can be reversed as easily as their insignificance can be exposed.

MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
References:
 [1] Huang, L. -G.; Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings, J. math. Anal. appl. 332, 1468-1476 (2007) · Zbl 1118.54022 · doi:10.1016/j.jmaa.2005.03.087 [2] Rezapour, Sh.; Hamlbarani, R.: Some notes on the paper ”cone metric spaces and fixed point theorems of contractive mappings”, J. math. Anal. appl. 345, 719-724 (2008) · Zbl 1145.54045 · doi:10.1016/j.jmaa.2008.04.049 [3] Abbas, M.; Jungck, G.: Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. math. Anal. appl. 341, 416-420 (2008) · Zbl 1147.54022 · doi:10.1016/j.jmaa.2007.09.070 [4] Ilić, D.; Rakočević, V.: Common fixed points for maps on a cone metric space, J. math. Anal. appl. 341, 876-882 (2008) · Zbl 1156.54023 · doi:10.1016/j.jmaa.2007.10.065 [5] Rezapour, Sh.; Haghi, R. H.; Shahzad, N.: Some notes on fixed points of quasi-contraction maps, Appl. math. Lett. 23, 498-502 (2010) · Zbl 1206.54061 · doi:10.1016/j.aml.2010.01.003 [6] Berinde, M.; Berinde, V.: On general class of multivalued weakly Picard mappings, J. math. Anal. appl. 326, 772-782 (2007) · Zbl 1117.47039 · doi:10.1016/j.jmaa.2006.03.016 [7] Kamran, T.: Multivalued f-weakly Picard mappings, Nonlinear anal. 67, 2289-2296 (2007) · Zbl 1128.54024 · doi:10.1016/j.na.2006.09.010 [8] Suzuki, T.: A new type of fixed point theorem in metric spaces, Nonlinear anal. 71, 5313-5317 (2009) · Zbl 1179.54071 · doi:10.1016/j.na.2009.04.017 [9] Edelstein, M.: On fixed and periodic points under contractive mappings, J. lond. Math. soc. 37, 74-79 (1962) · Zbl 0113.16503 · doi:10.1112/jlms/s1-37.1.74 [10] Singh, S. L.; Mishra, S. N.: Remarks on recent fixed point theorems, Fixed point theory appl. 2010 (2010) · Zbl 1204.47073 · doi:10.1155/2010/452905 [11] Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. math. Anal. appl. 340, 1088-1095 (2008) · Zbl 1140.47041 · doi:10.1016/j.jmaa.2007.09.023 [12] Browder, F. E.: Nonexpansive nonlinear operators in a Banach space, Proc. natl. Acad. sci. USA 54, 1041-1044 (1965) · Zbl 0128.35801 · doi:10.1073/pnas.54.4.1041 [13] Browder, F. B.: Fixed point theorems for noncompact mappings in Hilbert space, Proc. natl. Acad. sci. USA 53, 1272-1276 (1965) · Zbl 0125.35801 · doi:10.1073/pnas.53.6.1272 [14] Göhde, D.: Zum prinzip def kontractiven abbildung, Math. nachr. 30, 251-258 (1965)